Say you have a hollow sphere with a uniformly distributed charge on the surface. Why is the electric field everywhere inside the sphere zero?
For the centre, its easy to add the vectors from the surface charges and show they sum to zero because of symmetry.
But can you show me how the field cancels out to zero for points other than the centre using vector addition?
Answer
It is very simple, but strictly applies to a conducting sphere.
From ANY point inside a sphere, draw a double cone that is zero dimension at that point.
The cross-section of the cone, can be any shape.
The two ends of the cone intersect the sphere in two similar shaped curved surfaces. Any line from a point inside one of those areas, through the tip of the cone to the opposite surface has two sections with a ratio of A:B in length.
The areas of the two end cap surfaces, are also in the ratio of A^2:B^2 and so are the charges on those two areas.
Since the inverse distances squared are also A^2:B^2, the force on a charge at the cone tip is net zero. This is true for any point in the sphere, and any cone angle or cross-section shape.
QED
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